Euler angles equation

The Euler angles are often used to describe the orientations of a molecule. There are thre Euler cmgles d and ip. is a rotation about the Cartesian z axis this has the effec of moving the x and y axes, d is a rotation about the new x axis. Finally, ip is rotation about the new z axis (Figure 8.4). If the Euler angles are randomly changed b small amounts S[Pg.437]

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

Equation 7.2-22 indicates that the separating force is proportional to the local shear stress (fiy) in the liquid, it is a sensitive function of the Euler angles of orientation, and is proportional to the projection of the cross-sectional are (S = nc2). The angular velocities of rotation of the freely suspended spheroid particle were given by Zia, Cox, and Mason (46)... 

Note that, because Euler angles 0 and (f>, there is no contribution from the term in J2. Substituting (2.138) into (2.136), we obtain the wave equation for the rotation vibration wave functions in the Born adiabatic approximation ... 

We now consider how to evaluate the matrix elements of a spherical tensor operator, written as (rj, j, m Tk(A) rj, /". m ) where r] and if denote any further quantum numbers required to characterise the states (for example, vibrational quantum numbers v). If we now rotate the bra, operator and ket through Euler angles o> using equations (5.44) and (5.102), the result must be unaffected. Thus ... 

Each term on the right-hand side of this equation consists of the product of a direction cosine and an orbital angular momentum operator. Of these two factors, only the first depends on the Euler angles which define the instantaneous orientation of the molecule. In the corresponding equation (5.151) for./,, however, both factors depend on the Euler angles. 

Here, as usual, the suffices p and q refer to space- and molecule-fixed components respectively and > stands for the three Euler angles (. 0. /) which relate the two coordinate systems. From equations (5.155) and (5.156) we have... 

Thus we see that the operator g is not strictly an angular momentum operator in the quantum mechanical sense, which is why we have assigned it a different symbol. More importantly for the present purposes, we cannot use the armoury of angular momentum theory and spherical tensor methods to construct representations of the molecular Hamiltonian. In addition, the rotational kinetic energy operator, equation (7.89), takes a more complicated form than it has for a nonlinear molecule where there are three Euler angles (rotational coordinates). 

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

The analysis is not yet complete because we have to consider the left-hand term in equation (8.495) which relates the space-fixed and molecule-fixed coordinate systems. Although we have already selected the q = 0 component, we will retain q as a variable in order to facilitate later discussion. First we note that and can be expressed in terms of the Euler angles 0 and x as follows ... 

There are 3N + 7 coordinates on the right sides of Eq. (3.4), Le., the 3N vibrational displacements the three coordinates of the center of mass, the three Euler angles 0, x and the angle p. Since there are 3N coordinates/ /a (i = 1,2,.., N ot=x,y, z) on the left sides of Eq. (3.4), the 3N vibrational displacements are subject to seven constraint equations which further specify the molecule-fixed axis system. We shall use the following set of Eckart and Sayvetz conditions for these constraint equations ... 

From Eqs. (3.40) and (3.35) it is obvious that the inversion—rotation wave functions i//°. (0,, X, p) of NH3 which are the eigenfunctions of the operator, , can be written as a product of the rigid-rotor symmetric top wave functions depending on the Euler angles 0,4>, x and the inversion wave functions, depending on the variable p. Integration of the Schrodinger equation... 

This is the well-known Favro equation. The generators 7, can be defined in terms of the Euler angles 6, 9, and as follows ... 

Consider a molecule with orientation = 04, pp Yj. When we rotate this molecule over the Euler angles m, the set of Euler angles describing the new orientation of the molecule, may be obtained from the matrix equation ... 

This defines a set of equations for the mean field Hamiltonians HPF. These equations have to be solved self-consistently since the thermodynamic values within the angle brackets in (109) involve the mean field Hamiltonians // F. In principle, all // F can be different in practice, we impose symmetry relations. Therefore, we choose a unit cell, compatible with the symmetry of the lattice introduced in Section II,D, and we put Hpf equal to // F whenever P and P belong to the same sublattice. Moreover, we apply unit cell symmetry that relates the mean field Hamiltonians on different sublattices. By using the symmetry-adapted functions introduced in Section II,B, the latter symmetry can be imposed as follows. We select a set of molecules constituting the asymmetric part of the unit cell. Then we assign to all other molecules P Euler angles tip-through which the mean field. Hamiltonian of some molecule P in the asymmetric part has to be rotated in order to obtain HrF. As a result, we... 

Finally, the Hooke equations in a crystallite can be written by using the eomponents of the strain and stress tensors in the sample reference system. Denoting by g the triplet of Euler angles (q>i, Oq, q>2) and using Equations (64-66) we have ... 

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

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